# Particle-Dependent Deformations of Lorentz Symmetry

## Abstract

**:**

## 1. Introduction

## 2. DSR-Deformations of Lorentz Symmetry

## 3. Two Known Examples of DSR Setups with Universality

#### 3.1. A Setup with Commutative Composition of Momenta

#### 3.2. A $\kappa $-Poincaré Inspired Setup

## 4. Simplest Scenarios for a DSR Setup without Universality

#### 4.1. Simplest Case with Commutative Composition of Momenta

#### 4.2. Simplest $\kappa $-Poincaré-Inspired Case

## 5. A More General Scenario without Universality

## 6. Composite Particles and Potential Implications for Macroscopic Bodies

- (
**i**) - It could evidently be used to describe pictures in which different “elementary/fundamental” particles have different relativistic properties.
- (
**ii**) - It could also be used to describe pictures in which all “elementary/fundamental” particles have the same DSR-deformed relativistic properties, but “composite microscopic particles” such as atoms (because of the known mechanisms mentioned in Section I) have different relativistic properties, with weaker deformation of special-relativistic properties than the fundamental particles that compose them.
- (
**iii**) - And perhaps it could also be used to describe pictures in which microscopic particles have DSR-deformed relativistic properties, but macroscopic bodies (again because of the known mechanisms mentioned in Section I) have ordinary special-relativistic properties.

**i**)and (

**ii**), assuming indeed ${|\ell |}^{-1}$ is of the order of the Planck scale the new class of DSR-relativistic theories introduced in the previous two sections certainly provides plausible physical pictures, since for microscopic particles (even for atoms) the Planck scale is a gigantic scale and all effects of DSR deformation amount to small corrections.

**iii**) instead does not look too promising: Even for this case (

**iii**) the fact that I have shown here how different relativistic properties can coexist is a significant step forward, but for macroscopic bodies the Planck scale is actually a small energy scale and there is therefore the risk of predicting hugely unrealistic effects. It is also for this reason that so far the most popular way to handle macroscopic bodies in DSR research has been (see, e.g., [4,10]) the one of renouncing to the introduction of direct interactions between macroscopic and microscopic particles: The interaction with a macroscopic particle could be of course also described in terms of the microscopic interactions involving the constituents of the macroscopic body.

## 7. Implications for Particle-Physics Processes

**i**) in the previous section) this possible application is clearly viable if one assumes indeed that ${|\ell |}^{-1}$ is of the order of the Planck scale (or some other ultralarge momentum scale), since for microscopic particles (even for atoms) the Planck scale is gigantic and all effects of DSR deformation amount to small corrections.

#### 7.1. Absence of Cherenkov-like $Y\to Y+X$ Processes

#### 7.2. Absence of Anomalies for Particle Decays

#### 7.3. Aside on Quantum Gravity and the Planck Scale

## 8. Aside on “Hopf-Hopf Algebras with Mixing Co-Products”

## 9. Aside Beyond Leading Order

## 10. Closing Remarks

## References and Notes

- Amelino-Camelia, G. Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale. Int. J. Mod. Phys.
**2002**, D11, 35–60. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Testable scenario for Relativity with minimum-length. Phys. Lett.
**2001**, B510, 255–263. [Google Scholar] [CrossRef] - Kowalski-Glikman, J. Observer-independent quanta of mass and length. Phys. Lett.
**2001**, A286, 391–394. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Status of Relativity with observer-independent length and velocity scales. AIP Conf. Proc.
**2001**, 589, 137–150. [Google Scholar] - Magueijo, J.; Smolin, L. Generalized lorentz invariance with an invariant energy scale. Phys. Rev.
**2003**, D67. [Google Scholar] [CrossRef] - Kowalski-Glikman, J.; Nowak, S. Noncommutative space-time of doubly special relativity theories. Int. J. Mod. Phys.
**2003**, D12, 299–316. [Google Scholar] [CrossRef] - Magueijo, J.; Smolin, L. Gravity’s rainbow. Class. Quant. Grav.
**2004**, 21, 1725–1736. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Doubly-special relativity: Facts, myths and some key open issues. Symmetry
**2010**, 2, 230–271. [Google Scholar] [CrossRef] - Magueijo, J. Could quantum gravity be tested with high intensity lasers? Phys. Rev.
**2006**, D73. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. Relative locality and the soccer ball problem. Phys. Rev.
**2011**, D84, 087702:1–087702:4. [Google Scholar] [CrossRef] - Amelino-Camelia, G. Kinematical solution of the UHE-cosmic-ray puzzle without a preferred class of inertial observers. Int. J. Mod. Phys.
**2003**, D12, 1211–1226. [Google Scholar] [CrossRef] - Amelino-Camelia, G. On the fate of Lorentz symmetry in relative-locality momentum spaces. Phys. Rev.
**2012**, D85, 084034:1–084034:16. [Google Scholar] [CrossRef] - Gubitosi, G.; Mercati, F. Relative Locality in κ-Poincaré. Available online: http://arxiv.org/abs/1106.5710v1 (accessed on 27 June 2012).
- Amelino-Camelia, G.; Arzano, M.; Kowalski-Glikman, J.; Rosati, G.; Trevisan, G. Relative-locality distant observers and the phenomenology of momentum-space geometry. Class. Quant. Grav.
**2012**, 29. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. The principle of relative locality. Phys. Rev.
**2011**, D84, 084010:1–084010:12. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. Relative locality: A deepening of the relativity principle. Gen. Rel. Grav.
**2011**, 43, 2547–2553. [Google Scholar] [CrossRef] - Majid, S.; Ruegg, H. Bicrossproduct structure of kappa Poincaré group and noncommutative geometry. Phys. Lett. B
**1994**, 334, 348–354. [Google Scholar] [CrossRef] - Lukierski, J.; Ruegg, H.; Zakrzewski, W.J. Classical and quantum-mechanics of free κ-relativistic systems. Ann. Phys.
**1995**, 243, 90–116. [Google Scholar] [CrossRef] - Throughout this manuscript I write conservation laws at a process with conventions such that all momenta intervening in the process are incoming into the process, so that indeed a trivalent process would be characterized by a conservation law of the type $p{\oplus}_{\ell}{p}^{\prime}{\oplus}_{\ell}{p}^{\u2033}=0$. The case of one (or two) of the momenta that is outgoing from the process, say the momentum p, is recovered by simply substituting for p the “antipode" of the momentum of that outgoing particle, with the antipode $\ominus p$ defined so that $(\ominus p)\oplus p=0$. [For the composition law Equation (17) the antipode is such that ${\left({\ominus}_{\ell}p\right)}_{j}=-{p}_{j}+\ell {p}_{0}{p}_{j}$ and ${\left({\ominus}_{\ell}p\right)}_{0}=-{p}_{0}$.]
- Notice that in this specific example the limit of $p{\oplus}_{\ell \star}k$ as $p\to 0$ is not k. So scenarios such as this require a profound distinguishability between a single-particle system and a multi-particle system, a feature which is plausible but surprising. I should stress that this feature is not a general aspect of my “mixing composition laws", and indeed in the example Equation (30) one does find that $p{\oplus}_{\ell \star}k\to k$ as $p\to 0$. Also notice that in this subsection I could have opted for adopting as mixing composition law the limit $\lambda \to 0$ of the composition law Equation (62) from the next section, and in that case the limit $p\to 0$ does give k. As a matter of fact I opted for showing here Equation (46) rather than the limit $\lambda \to 0$ of Equation (62), just so that the content of this subsection would not simply amount to the $\lambda \to 0$ of the content of the next section, thereby showing some evidence of the variety of possibilities that could be considered (including the possibility of having $p{\oplus}_{\ell \star}k\ne k$ in the limit $p\to 0$).
- Adam, T.; Agafonova, N.; Aleksandrov, A.; Altinok, O.; Alvarez Sanchez, P.; Anokhina, A.; Aoki, S.; Ariga, A.; Ariga, T.; Autiero, D.; et al. Measurement of the neutrino velocity with the OPERA detector in the CNGS beam. Available online: http://arxiv.org/abs/1109.4897v2 (accessed on 27 June 2012).
- Cartlidge, E. Superluminal neutrinos loose cable may unravel faster-than-light result. Science
**2012**, 335. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Gubitosi, G.; Loret, N.; Mercati, F.; Rosati, G.; Lipari, P. OPERA-reassessing data on the energy dependence of the speed of neutrinos. Int. J. Mod. Phys.
**2011**, D20, 2623–2640. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. OPERA neutrinos and relativity. Available online: http://arxiv.org/abs/1110.0521v1 (accessed on 27 June 2012).
- Klinkhamer, F.R. Superluminal neutrino, flavor, and relativity. Phys. Rev. D
**2012**, 85. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Gubitosi, G.; Loret, N.; Mercati, F.; Rosati, G. Weakness of accelerator bounds on electron superluminality without a preferred frame. Available online: http://arxiv.org/abs/1111.0993v1 (accessed on 27 June 2012).
- Ling, Y. A note on superluminal neutrinos and deformed special relativity. Available online: http://arxiv.org/abs/1111.3716v1 (accessed on 27 June 2012).
- Huo, Y.; Li, T.; Liao, Y.; Nanopoulos, D.V.; Qi, Y.; Wang, F. The OPERA superluminal neutrinos from deformed lorentz invariance. Available online: http://arxiv.org/abs/1111.4994 (accessed on 27 June 2012).
- Aharonian, F.; Akhperjanian, A.G.; Barres de Almeida, U.; Bazer-Bachi, A.R.; Becherini, Y.; Behera, B.; Beilicke, M.; Benbow, W.; Bernlohr, K.; Boisson, C.; et al. Limits on an energy dependence of the speed of light from a flare of the active galaxy PKS 2155-304. Phys. Rev. Lett.
**2008**, 101, 170402:1–170402:5. [Google Scholar] [CrossRef] [PubMed] - Abdo, A.A.; Ackermann, M.; Arimoto, M.; Asano, K.; Atwood, W.B.; Axelsson, M.; Baldini, L.; Ballet, J.; Band, D.L.; Barbiellini, G.; et al. Fermi observations of high-energy gamma-ray emission from GRB 080916C. Science
**2009**, 323. [Google Scholar] [CrossRef] - Abdo, A.A.; Ackermann, M.; Ajello, M.; Asano, K.; Atwood, W.B.; Axelsson, M.; Baldini, L.; Ballet, J.; Barbiellini, G.; Baring, M.G.; et al. A limit on the variation of the speed of light arising from quantum gravity effects. Nature
**2009**, 462, 331–334. [Google Scholar] [CrossRef] [PubMed] - Amelino-Camelia, G.; D’Andrea, F.; Mandanici, G. Group velocity in noncommutative spacetime. JCAP
**2003**, 0309. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Matassa, M.; Mercati, F.; Rosati, G. Taming nonlocality in theories with deformed Poincare symmetry. Phys. Rev. Lett.
**2011**, 106. [Google Scholar] [CrossRef] [PubMed] - Smolin, L. On limitations of the extent of inertial frames in non-commutative relativistic spacetimes. Available online: http://arxiv.org/abs/1007.0718v3 (accessed on 27 June 2012).
- Amelino-Camelia, G.; Loret, N.; Rosati, G. Speed of particles and a relativity of locality in κ-Minkowski quantum spacetime. Phys. Lett.
**2011**, B700, 150–156. [Google Scholar] [CrossRef] - Cohen, A.G.; Glashow, S.L. New constraints on neutrino velocities. Phys. Rev. Lett.
**2011**, 107, 181803:1–181803:5. [Google Scholar] - Maccione, L.; Liberati, S.; Mattingly, D.M. Violations of Lorentz invariance in the neutrino sector after OPERA. Available online: http://arxiv.org/abs/1110.0783 (accessed on 27 June 2012).
- Carmona, J.M.; Cortes, J.L. Constraints from neutrino decay on superluminal velocities. Available online: http://arxiv.org/abs/1110.0430 (accessed on 27 June 2012).
- Amelino-Camelia, G. Improved limit on quantum-spacetime modifications of Lorentz symmetry from observations of gamma-ray blazars. New J. Phys.
**2004**, 6. [Google Scholar] [CrossRef] - Heyman, D.; Hinteleitner, F.; Major, S. On reaction thresholds in doubly special relativity. Phys. Rev.
**2004**, D69. [Google Scholar] [CrossRef] - Cowsik, R.; Nussinov, S.; Sarkar, U. Superluminal neutrinos at OPERA confront pion decay kinematics. Phys. Rev. Lett.
**2011**, 107. [Google Scholar] [CrossRef] - Gonzalez-Mestres, L. Astrophysical consequences of the OPERA superluminal neutrino. Available online: http://arxiv.org/abs/1109.6630 (accessed on 27 June 2012).
- Bi, X.-J.; Yin, P.-F.; Yu, Z.-H.; Yuan, Q. Constraints and tests of the superluminal neutrinos at OPERA. Phys. Rev. Lett.
**2011**, 107. [Google Scholar] [CrossRef] [PubMed] - Alfaro, J.; Morales-Tecotl, H.A.; Urrutia, L.F. Quantum gravity corrections to neutrino propagation. Phys. Rev. Lett.
**2000**, 84, 2318–2321. [Google Scholar] [CrossRef] [PubMed] - Alfaro, J. Quantum gravity and maximum attainable velocities in the standard model. AIP Conf. Proc.
**2007**, 917, 161–169. [Google Scholar] - Ellis, J.R.; Mavromatos, N.E.; Sakharov, A.S. Synchrotron radiation from the Crab Nebula discriminates between models of space-time foam. Astropart. Phys.
**2004**, 20, 669–682. [Google Scholar] [CrossRef] - Agostini, A.; Amelino-Camelia, G.; D’Andrea, F. Hopf-algebra description of noncommutative-spacetime symmetries. Int. J. Mod. Phys.
**2004**, A19. [Google Scholar] [CrossRef] - Note that a conservation law of the type ${p}_{0}+{k}_{0}=0,{p}_{j}+{k}_{j}+(\ell +\lambda ){p}_{0}{k}_{j}/2=0$ is equivalent to ${p}_{0}+{k}_{0}=0\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{p}_{j}+{k}_{j}+\lambda {p}_{0}{k}_{j}/2-\ell {k}_{0}{p}_{j}/2=0$ since (for ${p}_{0}+{k}_{0}=0$ and working at leading order) one has that from ${p}_{j}+{k}_{j}+\frac{\ell +\lambda}{2}{p}_{0}{k}_{j}=0$ it follows that $(1+\ell {k}_{0}/2)[{p}_{j}+{k}_{j}+\frac{\ell +\lambda}{2}{p}_{0}{k}_{j}]=0$ and in turn ${p}_{j}+{k}_{j}+\frac{\lambda}{2}{p}_{0}{k}_{j}-\frac{\ell}{2}{k}_{0}{p}_{j}=0$.

© 2012 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Amelino-Camelia, G.
Particle-Dependent Deformations of Lorentz Symmetry. *Symmetry* **2012**, *4*, 344-378.
https://doi.org/10.3390/sym4030344

**AMA Style**

Amelino-Camelia G.
Particle-Dependent Deformations of Lorentz Symmetry. *Symmetry*. 2012; 4(3):344-378.
https://doi.org/10.3390/sym4030344

**Chicago/Turabian Style**

Amelino-Camelia, Giovanni.
2012. "Particle-Dependent Deformations of Lorentz Symmetry" *Symmetry* 4, no. 3: 344-378.
https://doi.org/10.3390/sym4030344